
Like in the case of an elliptic modular form (see [1]}), the period function \(R_{G_{\mathbb {F},{k}}}(X) \)  is \(\frac{1}{\mathcal {N}(X)}\)  times a polynomial :
\(R_{G_{\mathbb {F},{k}}}(X) \in \frac{1}{\mathcal {N}(X)} \mathbb {C}[X]. \) 

 [2]} Note that
\(\frac{D^{n-\frac{1}{2}} \zeta _{\mathbb {F}}(n)\Gamma (n)^t}{(2\pi i)^{tn}}=\frac{\zeta _{\mathbb {F}}(1-n)}{2^t}\) 
and
\( \zeta _{\mathbb {F}}(-n)=0\)  for any positive even integer \(n.\)



[3]} Let
\(\mathbb {F}\)  be a real quadratic field with discriminant \(D.\) 
It is known that
\(\zeta _{\mathbb {F}}(1-n)=B_{n}B_{n,\chi }\)  for even positive integer \(n.\) 
\(B_r\)  and \( B_{r,\chi }\)  are
the \(r\) th Bernoulli number \((B_0=1, B_1=-\frac{1}{2}, B_2=\frac{1}{6}, \cdots )\)  and the \(r\) th twisted Bernoulli number \(( B_{1,\chi }=\frac{1}{D}\sum _{a=1}^D \chi (a)a, B_{2,\chi }=\frac{1}{D}\sum _{a=1}^{D}\chi (a)a^2-\sum _{a=1}^D \chi (a) a, B_{3,\chi }= \cdots ),\)  respectively. Here
\(\chi \pmod {D} \)  is a primitive character defined by \(\chi (\cdot )=\bigl (\frac{D}{\cdot }\bigr ). \)

 (open problem) 
It is well known that the generating functions of \(B_j\)  and \(B_{j,\chi } \)  are
\(\sum _{n=0}^{\infty }B_n\frac{t^n}{n!}=\frac{te^t}{e^t-1}\mbox{\, and } \sum _{m\ge 0} B_{m,\chi }\frac{t^m}{m!} = \sum _{j=1}^{D}\frac{\chi (j) t e^{jt}}{e^{Dt}-1}.\) 
Similarly, it will be interesting
to express the following generating function
\( \sum _{m\ge 2}^{\infty } B_m B_{m, \chi }\frac{t^m}{m!}=\sum _{m\ge 2}^{\infty }\zeta _{\mathbb {F}}(1-m)\frac{t^m}{m!},\) 
as elementary functions.



